Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 133 / 2 / a / i

Write

G = ⟨ u ⟩ *_{⟨ u^{2} ⟩ = ⟨ v^{3} ⟩} ⟨ v ⟩ .

G / ⟨ u ⟩ ⊔ G / ⟨ v ⟩

and edges

G / C

, where

C = ⟨ u^{2} ⟩ = ⟨ v^{3} ⟩

; the edge

g C

joins

g ⟨ u ⟩

g ⟨ v ⟩

. Since

[⟨ u ⟩ : ⟨ u^{2} ⟩] = 2, [⟨ v ⟩ : ⟨ v^{3} ⟩] = 3,

this is the infinite

(2, 3)

-biregular tree: every

u

-type vertex has degree two and every

v

-type vertex has degree three.

Solved by gpt-5.6-sol high.

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